Isomorphism Of Braid Groups And Cyclotomic Hecke Algebras Corresponding To A Complex Reflection Group

The classification of the presentations into congruence ones has been completed for themost complex reflection groups(see[1],[2],[6],[7]).Each presentation (S,P) of a complexgroup G gives rise to a braid group G_S,P and a cyclotomic Hecke algebra.A question is to askwhether the braid groups corresponding to various presentations of G are isomorphic or not.The same question can be asked for the corresponding cyclotomic Hecke algebra.For eachcomplex reflection groups G₇,G₁₁,G₁₂,G₁₅,G₂₄,G₂₅,G₂₆,G₂₇,G₃₂,we have proved braidgroups corresponding to its some presentations are isomorphic.Furthermore,for presentations of complex reflection groups G₁₂,G₁₅,G₂₄,G₂₆,G₂₇,G₃₂,we have proved cyclotomicHecke algebra corresponding to each of them are isomorphic.We use homomorphism basictheorem in the proofs,and in a few of proofs we use mathematical software GAP.The necessary commands in GAP are listed in the appendix.